L(s) = 1 | + (−0.617 + 0.786i)2-s + (−0.859 − 0.511i)3-s + (−0.238 − 0.971i)4-s + (0.932 − 0.360i)6-s + (−0.848 − 0.529i)7-s + (0.911 + 0.412i)8-s + (0.476 + 0.879i)9-s + (0.895 + 0.444i)11-s + (−0.292 + 0.956i)12-s + (0.826 − 0.563i)13-s + (0.940 − 0.340i)14-s + (−0.886 + 0.462i)16-s + (−0.984 − 0.178i)17-s + (−0.985 − 0.168i)18-s + (0.992 + 0.122i)19-s + ⋯ |
L(s) = 1 | + (−0.617 + 0.786i)2-s + (−0.859 − 0.511i)3-s + (−0.238 − 0.971i)4-s + (0.932 − 0.360i)6-s + (−0.848 − 0.529i)7-s + (0.911 + 0.412i)8-s + (0.476 + 0.879i)9-s + (0.895 + 0.444i)11-s + (−0.292 + 0.956i)12-s + (0.826 − 0.563i)13-s + (0.940 − 0.340i)14-s + (−0.886 + 0.462i)16-s + (−0.984 − 0.178i)17-s + (−0.985 − 0.168i)18-s + (0.992 + 0.122i)19-s + ⋯ |
Λ(s)=(=(6145s/2ΓR(s+1)L(s)(−0.139+0.990i)Λ(1−s)
Λ(s)=(=(6145s/2ΓR(s+1)L(s)(−0.139+0.990i)Λ(1−s)
Degree: |
1 |
Conductor: |
6145
= 5⋅1229
|
Sign: |
−0.139+0.990i
|
Analytic conductor: |
660.371 |
Root analytic conductor: |
660.371 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ6145(112,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 6145, (1: ), −0.139+0.990i)
|
Particular Values
L(21) |
≈ |
0.3188412047+0.3667441261i |
L(21) |
≈ |
0.3188412047+0.3667441261i |
L(1) |
≈ |
0.5316084831+0.06003288065i |
L(1) |
≈ |
0.5316084831+0.06003288065i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 1229 | 1 |
good | 2 | 1+(−0.617+0.786i)T |
| 3 | 1+(−0.859−0.511i)T |
| 7 | 1+(−0.848−0.529i)T |
| 11 | 1+(0.895+0.444i)T |
| 13 | 1+(0.826−0.563i)T |
| 17 | 1+(−0.984−0.178i)T |
| 19 | 1+(0.992+0.122i)T |
| 23 | 1+(−0.262+0.964i)T |
| 29 | 1+(−0.267+0.963i)T |
| 31 | 1+(−0.997+0.0715i)T |
| 37 | 1+(−0.542−0.840i)T |
| 41 | 1+(0.257−0.966i)T |
| 43 | 1+(−0.102−0.994i)T |
| 47 | 1+(−0.471+0.881i)T |
| 53 | 1+(−0.874−0.485i)T |
| 59 | 1+(−0.316−0.948i)T |
| 61 | 1+(0.355+0.934i)T |
| 67 | 1+(0.995+0.0970i)T |
| 71 | 1+(0.811−0.584i)T |
| 73 | 1+(−0.993−0.117i)T |
| 79 | 1+(−0.412−0.911i)T |
| 83 | 1+(0.902+0.430i)T |
| 89 | 1+(−0.502+0.864i)T |
| 97 | 1+(−0.708−0.705i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−17.28960404945270653494656184204, −16.7432328261993722558879486101, −16.149075501203762003627431177402, −15.783164444862716838209786518561, −14.852781268536110971843169516131, −13.844916720169518154213061945149, −13.17152455026543043647120709739, −12.53340290858164192301667512242, −11.819035145908835340169454094573, −11.33148846443576596578171476127, −10.92109875061256624496643620613, −9.87289826414094611350708807046, −9.57764019030100845315928475854, −8.87728836202056384493894480102, −8.3471727030478056249234367807, −7.115431033564388231073634824470, −6.45885195091855685891647098748, −6.03156799903094476733160584901, −4.928296047526315639605139509906, −4.14048505912964493531973578515, −3.60759388039864053170759170835, −2.87828828158501801664914880916, −1.82571352057156616739922343920, −1.02356240101523553528259066847, −0.17292227733055699202221612669,
0.55118987845516665875860962346, 1.350104307308432741280074583303, 1.970070981788753322495566837004, 3.43137829047396332527436890038, 4.11449132750444796119879136132, 5.13882415536515094913888696924, 5.68080332935908459292041221218, 6.367628427480199399700891873360, 7.05153621460056972161937124060, 7.295913197994477803106705690216, 8.19712820420642361280045909153, 9.21102022893066886528764237766, 9.505790926391438988503809184782, 10.514960463867944534659297262962, 10.89492215610008961896958294954, 11.63471087428712945524509205467, 12.545755987483435851061802823076, 13.15263919940054327389035459419, 13.80515232582336969800844408197, 14.34671960495216765280887850065, 15.456808118348257969057433355670, 15.95422660356030296409239090124, 16.33502108248296329685666899359, 17.14336372532997795327607754175, 17.69980564620499015775483743123